Theorems · Theorem · ring theory
IsSemisimpleRing.exists_algEquiv_pi_matrix_of_isAlgClosed
∀ (F : Type u_1) (R : Type u_2) [inst : Field F] [IsAlgClosed F] [inst_2 : Ring R] [inst_3 : Algebra F R] [IsSemisimpleRing R] [FiniteDimensional F R], ∃ n d, (∀ (i : Fin n), NeZero (d i)) ∧ Nonempty (R ≃ₐ[F] (i : Fin n) → Matrix (Fin (d i)) (Fin (d i)) F)
The Wedderburn–Artin Theorem over algebraically closed fields: a finite-dimensional semisimple algebra over an algebraically closed field is isomorphic to a product of matrix algebras over the field.
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- Foundations
- Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound
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- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- Fieldstatement and proof · cited by 7,404
- Matrixstatement and proof · cited by 4,303
- FiniteDimensionalstatement and proof · cited by 1,854
- AlgEquivstatement and proof · cited by 1,681
- DivisionRingproof · cited by 1,062
- Module.Finiteproof · cited by 1,032
- AlgEquiv.symmproof · cited by 615
- Algebra.ofIdproof · cited by 166
- IsAlgClosedstatement and proof · cited by 150
- AlgEquiv.transproof · cited by 108
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